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A new type of superorthogonality

Philip T. Gressman, Lillian B. Pierce, Joris Roos, Po-Lam Yung

TL;DR

The paper introduces Type IV superorthogonality, a weak yet powerful vanishing condition for 2r-fold products of a function family that yields unconditional square function estimates on L^{2r} for all even exponents. A central technical tool is a pointwise inequality for Q_k comparing mixed-product sums to the product of sums, which, combined with a meticulous induction, yields explicit constants C_r ≤ √((2r)!,−1) and the decoupling form ||∑ f_j||_{L^{2r}} ≤ C_r (∑ ||f_j||_{L^{2r}}^2)^{1/2}. The authors prove Type IV implies the desired square function bounds, supply concrete Type IV examples (including Haar functions) and show that dyadic subfamilies recover the classical dyadic square function estimates. They also discuss a hierarchy of superorthogonality types, potential improvements of constants, and possible extensions to non-archimedean settings and almost- or quasi-superorthogonality concepts. This framework unifies and extends a broad class of square function results by reducing verification to a minimal vanishing criterion on index tuples.

Abstract

We provide a simple criterion on a family of functions that implies a square function estimate on $L^p$ for every even integer $p \geq 2$. This defines a new type of superorthogonality that is verified by checking a less restrictive criterion than any other type of superorthogonality that is currently known.

A new type of superorthogonality

TL;DR

The paper introduces Type IV superorthogonality, a weak yet powerful vanishing condition for 2r-fold products of a function family that yields unconditional square function estimates on L^{2r} for all even exponents. A central technical tool is a pointwise inequality for Q_k comparing mixed-product sums to the product of sums, which, combined with a meticulous induction, yields explicit constants C_r ≤ √((2r)!,−1) and the decoupling form ||∑ f_j||_{L^{2r}} ≤ C_r (∑ ||f_j||_{L^{2r}}^2)^{1/2}. The authors prove Type IV implies the desired square function bounds, supply concrete Type IV examples (including Haar functions) and show that dyadic subfamilies recover the classical dyadic square function estimates. They also discuss a hierarchy of superorthogonality types, potential improvements of constants, and possible extensions to non-archimedean settings and almost- or quasi-superorthogonality concepts. This framework unifies and extends a broad class of square function results by reducing verification to a minimal vanishing criterion on index tuples.

Abstract

We provide a simple criterion on a family of functions that implies a square function estimate on for every even integer . This defines a new type of superorthogonality that is verified by checking a less restrictive criterion than any other type of superorthogonality that is currently known.
Paper Structure (9 sections, 4 theorems, 41 equations)

This paper contains 9 sections, 4 theorems, 41 equations.

Table of Contents

  1. Statement of the results
  2. Square function estimate
  3. Superorthogonality
  4. Type IV superorthogonality
  5. Proof of Proposition \ref{['prop_numerical']}
  6. Proof of Theorem \ref{['thm_main']}
  7. Construction of an example: proof of Theorem \ref{['thm_example']}
  8. Haar functions: Proposition \ref{['prop_Haar']}
  9. Further directions

Key Result

Theorem 1

Let $r \geq 1$ be an integer. Suppose a family $\{f_j\}_{j \in J}$ of functions has the property that whenever $j_1,\ldots,j_{2r}$ are all distinct. If $(\sum_{j\in J} |f_j|^2 )^\frac{1}{2}$ belongs to $L^{2r}$, the series $\sum_{j \in J} f_j$ converges unconditionally in $L^{2r}$, and In particular, we may take $C_r=1$ for $r=1$ and $C_r \leq 2^{1/2}((2r)!-1)^{1/2}$ for $r \geq 2$.

Theorems & Definitions (4)

  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Proposition 4